For each quantum system described by an operator algebra [Formula: see text] of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen-Specker theorem. This quantum state space is the spectral presheaf [Formula: see text]. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of t...

Suppose that S is a space. There is an injective and a projective model structure for the resulting category of spaces with S-action, and both are easily derived. These model structures are special cases of model structures for presheaf-valued diagrams X defined on a fixed presheaf of categories E which is enriched in simplicial sets. Varying the parameter category object E (or parameter space ...

For a broad collection of categories K, including all presheaf categories, the following statement is proved to be consistent: every left exact (i.e. finite-limits preserving) functor from K to Set is small, that is, a small colimit of representables. In contrast, for the (presheaf) category K = Alg(1, 1) of unary algebras we construct a functor from Alg(1, 1) to Set which preserves finite prod...

Abstract We prove a correspondence between $\kappa$ -small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively -compact maps their underlying quasi-categories for suitably large regular cardinals . thus obtain transition result weakly universal small (type-theoretic) Dugger–Rezk-style st...

Process calculi like CCS have been motivated and studied operationally, thus from the outset lacking the abstract mathematical treatment provided by a domain theory. Consequently, concurrency has become a rather separate study; in particular, higher-order and functional features as known from sequential programming are most often treated in an ad hoc fashion, if at all. The study of presheaf mo...

This paper introduces new-HOPLA, a concise but powerful language for higher-order nondeterministic processes with name generation. Its origins as a metalan-guage for domain theory are sketched but for the most part the paper concentrateson its operational semantics. The language is typed, the type of a process describingthe shape of the computation paths it can perform. Its ...

Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory.

A metalanguage for concurrent process languages is introduced. Within it a range of process languages can be defined, including higher-order process languages where processes are passed and received as arguments. (The process language has, however, to be linear, in the sense that a process received as an argument can be run at most once, and not include name generation as in the Pi-Calculus.) T...